Problem Analysis #0

Gemini-3-Pro-Preview · Run 1

316.27s

12,238 toks

Problem Statement

Consider the curve $C$ given in polar coordinates by

r = 2 + \cos\theta \qquad (0 \le \theta \le 2\pi),

with the origin $O$ as the pole and the positive $x$ –axis as the initial line.

[(1)] Compute the area of the region enclosed by the curve $C$ .
[(2)] At the point

(r,\theta)=\left(\frac{4+\sqrt{2}}{2},\,\frac{\pi}{4}\right),

find the slope of the tangent line to the curve $C$ in the $xy$–coordinate plane.

Let $a$ be the value obtained in part (1). Express the answer from part (2) in the form

b + \sqrt{c},

where $b$ and $c$ are integers, and let those integers be the definitions of $b$ and $c$ . Using these values of $a$ , $b$ , and $c$ , compute

\frac{abc}{\pi}.

Ground Truth

-18

Execution Trace

Reasoning Process

[PYTHON] import sympy as sp

Define the variable theta

theta = sp.symbols('theta')

Define the polar curve r

r = 2 + sp.cos(theta)

Part (1): Compute the area a

Area = (1/2) * integral(r^2, theta, 0, 2*pi)

area_expr = (1/2) * r**2 a = sp.integrate(area_expr, (theta, 0, 2*sp.pi)) print(f"Area a: {a}")

Part (2): Compute the slope at theta = pi/4

x = r * cos(theta), y = r * sin(theta)

x = r * sp.cos(theta) y = r * sp.sin(theta)

Derivatives with respect to theta

dx_dtheta = sp.diff(x, theta) dy_dtheta = sp.diff(y, theta)

Slope dy/dx = (dy/dtheta) / (dx/dtheta)

slope_expr = dy_dtheta / dx_dtheta

Evaluate at theta = pi/4

theta_val = sp.pi / 4 slope_val = slope_expr.subs(theta, theta_val) print(f"Slope value: {slope_val}") slope_simplified = sp.simplify(slope_val) print(f"Simplified slope: {slope_simplified}")

[/PYTHON] [PYTHON] import sympy as sp

theta = sp.symbols('theta') r = 2 + sp.cos(theta)

Area

a = sp.integrate((1/2) * r**2, (theta, 0, 2*sp.pi)) print(f"a = {a}")

Slope

x = r * sp.cos(theta) y = r * sp.sin(theta) dx = sp.diff(x, theta) dy = sp.diff(y, theta) slope = dy/dx slope_at_pi_4 = slope.subs(theta, sp.pi/4) simplified_slope = sp.simplify(slope_at_pi_4) print(f"Slope at pi/4 = {simplified_slope}")

Extract b and c

We expect form b + sqrt(c)

Let's try to match it

simplified_slope should be -2 + sqrt(2)

b = -2 c = 2 val = b + sp.sqrt(c) check = sp.simplify(simplified_slope - val) print(f"Check if slope is -2 + sqrt(2): {check == 0}")

Final calculation

result = (a * b * c) / sp.pi print(f"Result = {result}") [/PYTHON] [PYTHON] import sympy as sp

Define symbols

theta = sp.symbols('theta') r = 2 + sp.cos(theta)

Part 1: Area

Area = 1/2 * integral(r^2 dtheta) from 0 to 2pi

area = sp.integrate((1/2) * r**2, (theta, 0, 2*sp.pi)) print(f"Area a = {area}")

Part 2: Slope

x = r cos(theta), y = r sin(theta)

x = r * sp.cos(theta) y = r * sp.sin(theta)

dy/dx = (dy/dtheta) / (dx/dtheta)

dy_dtheta = sp.diff(y, theta) dx_dtheta = sp.diff(x, theta) slope_expr = dy_dtheta / dx_dtheta

Evaluate at theta = pi/4

slope_val = slope_expr.subs(theta, sp.pi/4) slope_simplified = sp.simplify(slope_val) print(f"Slope at pi/4 = {slope_simplified}")

Check form b + sqrt(c)

We suspect b = -2, c = 2

b = -2 c = 2 target = b + sp.sqrt(c) if sp.simplify(slope_simplified - target) == 0: print(f"Confirmed: Slope is {b} + sqrt({c})") else: print("Mismatch in slope form")

Compute final answer

final_val = (area * b * c) / sp.pi print(f"Final value abc/pi = {final_val}") [/PYTHON] The curve $C$ is given by $r = 2 + \cos\theta$ for $0 \le \theta \le 2\pi$ .

(1) Area of the enclosed region: The area $a$ is given by the integral:

a = \frac{1}{2} \int_{0}^{2\pi} r^2 \, d\theta = \frac{1}{2} \int_{0}^{2\pi} (2 + \cos\theta)^2 \, d\theta

Expanding the integrand:

(2 + \cos\theta)^2 = 4 + 4\cos\theta + \cos^2\theta

Integrating term by term:

\int_{0}^{2\pi} 4 \, d\theta = 8\pi

\int_{0}^{2\pi} 4\cos\theta \, d\theta = 0

\int_{0}^{2\pi} \cos^2\theta \, d\theta = \int_{0}^{2\pi} \frac{1 + \cos(2\theta)}{2} \, d\theta = \left[ \frac{\theta}{2} + \frac{\sin(2\theta)}{4} \right]_{0}^{2\pi} = \pi

Thus, the integral is $8\pi + 0 + \pi = 9\pi$ . The area is:

a = \frac{1}{2}(9\pi) = \frac{9\pi}{2}

(2) Slope of the tangent line: We need the slope at $(r, \theta) = \left(\frac{4+\sqrt{2}}{2}, \frac{\pi}{4}\right)$ . The parametric equations are:

x = r\cos\theta = (2+\cos\theta)\cos\theta = 2\cos\theta + \cos^2\theta

y = r\sin\theta = (2+\cos\theta)\sin\theta = 2\sin\theta + \sin\theta\cos\theta = 2\sin\theta + \frac{1}{2}\sin(2\theta)

Differentiating with respect to $\theta$ :

\frac{dx}{d\theta} = -2\sin\theta - 2\cos\theta\sin\theta = -2\sin\theta - \sin(2\theta)

\frac{dy}{d\theta} = 2\cos\theta + \cos(2\theta)

At $\theta = \frac{\pi}{4}$ :

\frac{dx}{d\theta} = -2\left(\frac{\sqrt{2}}{2}\right) - \sin\left(\frac{\pi}{2}\right) = -\sqrt{2} - 1

\frac{dy}{d\theta} = 2\left(\frac{\sqrt{2}}{2}\right) + \cos\left(\frac{\pi}{2}\right) = \sqrt{2} + 0 = \sqrt{2}

The slope $m$ is:

m = \frac{dy/d\theta}{dx/d\theta} = \frac{\sqrt{2}}{-\sqrt{2}-1} = \frac{\sqrt{2}}{-(1+\sqrt{2})}

Rationalizing the denominator:

m = \frac{\sqrt{2}(\sqrt{2}-1)}{-(1+\sqrt{2})(\sqrt{2}-1)} = \frac{2-\sqrt{2}}{-(2-1)} = \frac{2-\sqrt{2}}{-1} = -2 + \sqrt{2}

This is in the form $b + \sqrt{c}$ with integers $b = -2$ and $c = 2$ .

Final Calculation: We have $a = \frac{9\pi}{2}$ , $b = -2$ , and $c = 2$ . We need to compute $\frac{abc}{\pi}$ :

\frac{abc}{\pi} = \frac{\left(\frac{9\pi}{2}\right)(-2)(2)}{\pi} = \frac{-18\pi}{\pi} = -18

[THOUGHT] The calculations are verified. Area $a = 9\pi/2$ . Slope $m = -2 + \sqrt{2}$ . $b = -2, c = 2$ . Expression $\frac{abc}{\pi} = -18$ . [/THOUGHT]

$\boxed{-18}$

Tool Output

[PYTHON OUTPUT] Area a: 4.5*pi Slope value: (-1/2 + sqrt(2)*(sqrt(2)/2 + 2)/2)/(-sqrt(2)*(sqrt(2)/2 + 2)/2 - 1/2) Simplified slope: -2 + sqrt(2) [/PYTHON OUTPUT] [PYTHON OUTPUT] a = 4.5*pi Slope at pi/4 = -2 + sqrt(2) Check if slope is -2 + sqrt(2): True Result = -18.0000000000000 [/PYTHON OUTPUT] [PYTHON OUTPUT] Area a = 4.5*pi Slope at pi/4 = -2 + sqrt(2) Confirmed: Slope is -2 + sqrt(2) Final value abc/pi = -18.0000000000000 [/PYTHON OUTPUT]

Reasoning Process

$\boxed{-18}$

Final Model Output

-18

Comparative Analysis (P0) (Average across runs)

Avg Time Taken

Gemini-3-Pro-Preview

316.3

K-EXAONE-236B-A23B

234.2

Deepseek-V3.2

81.3

Kanana-2-30B-Thinking

70.7

Kanana-2-30B-Thinking-2601

59.2

Llama-VARCO-8B-Instruct

49.3

GPT-5.1 (high)

38.4

HCX-007(high)

36.4

GPT-5.2 (high)

33.6

Solar-Pro-2 (31B)(high)

30.2

K-EXAONE-236B-A23B

21.4

Grok-4.1-fast

21.1

Claude-Opus-4.5

18.0

Solar-Open-100B

17.0

A.X-4.0 (72B)

16.6

EXAONE-4.0.1-32B (high)

6.7

Avg Token Usage

K-EXAONE-236B-A23B

22,901

Gemini-3-Pro-Preview

12,238

Kanana-2-30B-Thinking-2601

12,235

Kanana-2-30B-Thinking

12,196

Deepseek-V3.2

7,244

GPT-5.2 (high)

7,100

Solar-Pro-2 (31B)(high)

6,632

K-EXAONE-236B-A23B

6,596

HCX-007(high)

6,397

Grok-4.1-fast

6,290

Solar-Open-100B

5,244

GPT-5.1 (high)

4,822

Claude-Opus-4.5

4,528

Llama-VARCO-8B-Instruct

4,316

EXAONE-4.0.1-32B (high)

4,306

A.X-4.0 (72B)

3,605